充分条件

On the basic that existing second-order sufficient condition for an isolated minimizer of order 2 for unconstrained multi-objective optimization, using the so called oriented distance from a point to a set, l-stable function and generalized second-order Peano direction derivative, some second-order sufficient conditions for an isolated minimizer of order 2 for constrained multiobjective optimization problems with l-stable functions are obtained.

在已有的无约束l-稳定函数多目标优化问题的二阶充分条件的基础上，借助定向距离函数和l-稳定函数的性质及引入的广义二阶Peano方向导数，进一步刻画了具有抽象约束的l-稳定函数的多目标优化问题的二阶孤立极小点的二阶充分条件。

Based on this reformulation, we first give the conditions under which the stationary point of unconstrained optimization problem is a solution of GNCP, and then give the conditions under which Hessian matrix of vector-valued function is nonsingular.

基于此转化形式，我们首先研究优化问题的稳定点为GNCP问题解的充分条件，然后建立无约束优化问题的向量价值函数的Hessian矩阵非奇异的充分条件。

Then it quotes a sufficient condition, the necessary conditions and necessary and sufficient conditions for integration, including the application of mathematical analysis, Real Variable Function-related content which is useful to sum up the necessary and sufficient conditions of the integrability.

然后引述了积分的充分条件、必要条件及充分必要条件，包括应用数学分析、实变函数相关内容总结的可积的充要条件，并作了翔实扼要的论述和证明，且给出针对性极强的应用例题，以便加深对可积条件的理解及加强应用能力。

This paper makes use of a Lyapunov function and a nonlinear integral inequality of Gronwall-Bellman-Bihari type to study the asymptotic hehaviors of solutions of a second-order nonlinear functional differential equation.

本文利用李雅普诺夫函数及一个Gronwall—Bellman—Bihari型的非线性积分不等式，研究一类二阶非线性泛函微分方程解的渐近性态，在第一部份，给出一组方程的全体解可延展到t=+∞的充分条件及三组方程的全体解有界的充分条件。

This paper makes use of a Lyapunov function and a nonlinear integral inequality of Gronwall-Bellman-Bihari type to study the asymptotic hehaviors of solutions of a second-order nonlinear functional differential equation. In part I, a group of sufficient conditions is given on continuation of solutions and three groups of sufficient conditions are given on boundedness of solutuicns of equation.

本文利用李雅普诺夫函数及一个Gronwall—Bellman—Bihari型的非线性积分不等式，研究一类二阶非线性泛函微分方程解的渐近性态，在第一部份，给出一组方程的全体解可延展到t=+∞的充分条件及三组方程的全体解有界的充分条件。

In this paper, we give a new sufficient condition and describe some semicomplete multipartite digraphs demonstrating that this sufficient condition is independent and in some sense, it is best posssible.

本文将针对此问题给出一个新的充分条件，并举例说明该充分条件的独立性以及它在某种意义下的最佳可能性。

In this thesis, some theories and approaches related to continuous dynamics, discrete dynamics, impulsive dynamics, operator theory and numerical simulations are used to investigate dynamical behaviors including the existence and globally asymptotic stability of periodic solutions, permanence, extinction, and all kinds of complexities, and meanwhile the possible effects of delay and impulse on the dynamical behaviors are discussed.

我们的这些结果显示出这三个时滞周期系统正周期解存在且全局渐近稳定的充分条件在形式上几乎能够很好的与相应的无时滞自治系统正平衡态存在且全局渐近稳定的充分条件相对(来源：A5babBC论文网www.abclunwen.com)应，这些结果也显示了时滞对系统正周期解的存在性和全局渐近稳定性是有影响的，它们推广并改进了一些已有的相关结果。

In chapter three, we study the lower orientable strong radius and strong diameterof the Cartesian product of graphs and prove that: srad(G_1×G_2)= 2r(G_1×G_2),sdiam(G_1×G_2)≤min{sdiam(G_1)+sdiam(G_2), 2(G_1×G_2), 4r(G_1×G_2)}. Furthermore,we establish three sufficient conditions for sdiam(G_1×G_2)= 2d(G_1×G_2)holds and determine the values of the lower orientable strong diameters of somespecial graphs. Moreover, we give the exact value of SDIAM, a lowerbound for SDIAM, an upper and lower bound for SRAD andSRAD, respectively.

在第三章，研究了笛卡尔乘积图G_1×G_2的最小强半径，证明了如下结果：srad(G_1×G_2)=2r(G_1×G_2)，sdiam(G_1×G_2)≤min{sdiam(G_1)+sdiam(G_2)，2d(G_1×G_2)，4r(G_1×G_2)；给出sdiam(G_1×G_2)=2d(G_1×G_2)成立的三个充分条件，并由所给出的充分条件确定了一些特殊笛卡尔乘积图的最小强直径的值；确定了SDIAM的确切值，SDIAM的下界，SRAD和SRAD的上、下界。

Using Taylor Theorem,we generalize the second sufficient conditions for extreme point and inflection point,and give the classification for a large class of stationary point .

利用泰勒定理，推广了极值的第二充分条件和拐点的第二充分条件，并对某一大类驻点进行了分类。

According to the model's actual background,discussing the random variable coefficient equation and using direct method,we obtain sufficient condition for the uniform stability and the globally asymptotic stability of difference equation,and also give examples to show they can not substitute each other.

本文根据种群模型的实际背景，讨论了任意变系数方程利用直接方法获得了所讨论方程的零解的一致稳定和全局渐近稳定的充分条件，并举例说明了这两个充分条件不能相互代替。

On the other hand, the more important thing is because the urban housing is a kind of heterogeneity products.

另一方面，更重要的是由于城市住房是一种异质性产品。

Climate histogram is the fall that collects place measure calm value, cent serves as cross axle for a few equal interval, the area that the frequency that the value appears according to place is accumulated and becomes will be determined inside each interval, discharge the graph that rise with post, also be called histogram.

气候直方图是将所收集的降水量测定值，分为几个相等的区间作为横轴，并将各区间内所测定值依所出现的次数累积而成的面积，用柱子排起来的图形，也叫做柱状图。